Monte Carlo Simulation

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1. Virtual Reality Goggle Inventory. Galaxy Co. sells virtual reality (VR)
goggles, particularly targeting customers who like to play video
games. Galaxy procures each pair of goggles for $150 from its
supplier and sells each pair of goggles for $300. Monthly demand for
the VR goggles is a normal random variable with a mean of 160 units
and a standard deviation of 40 units. At the beginning of each month,
Galaxy orders enough goggles from its supplier to bring the inventory
level up to 140 goggles. If the monthly demand is less than 140, Galaxy
pays $20 per pair of goggles that remains in inventory at the end of
the month. If the monthly demand exceeds 140, Galaxy sells only the
140 pairs of goggles in stock. Galaxy assigns a shortage cost of $40 for
each unit of demand that is unsatisfied to represent a loss-of-goodwill
among its customers. Management would like to use a simulation
model to analyze this situation.
a. What is the average monthly profit resulting from its policy of
stocking 140 pairs of goggles at the beginning of each month?
b. What is the proportion of months in which demand is
completely satisfied?
c. Use the simulation model to compare the profitability of
monthly replenishment levels of 140 and 160 pairs of goggles.
Use a 95% confidence interval on the difference between the
average profit that each replenishment level generates to make
your comparison.
2. Dice Rolls. Construct a spreadsheet simulation model to simulate
1,000 rolls of a die with the six sides numbered 1, 2, 3, 4, 5, and 6.
a. Construct a histogram of the 1,000 observed dice rolls.
SHOW ANSWER
b. For each roll of two dice, record the sum of the dice. Construct a
histogram of the 1,000 observations of the sum of two dice.
SHOW ANSWER
c. For each roll of three dice, record the sum of the dice. Construct
a histogram of the 1,000 observations of the sum of three dice.
SHOW ANSWER
d. For each roll of four dice, record the sum of the dice. Construct a
histogram of the 1,000 observations of the sum of four dice.
SHOW ANSWER
e. Compare the histograms in parts (a), (b), (c), and (d). What
statistical phenomenon does this sequence of charts illustrate?

 

Sample Solution

. Virtual Reality Goggle Inventory

a. Average Monthly Profit

To calculate the average monthly profit, we need to consider the following factors:

  • Revenue from sales: Each pair of goggles sold generates a revenue of $300.

  • Procurement cost: Each pair of goggles costs $150 to procure.

  • Inventory holding cost: If there are any unsold goggles at the end of the month, Galaxy incurs an inventory holding cost of $20 per pair.

  • Shortage cost: If Galaxy is unable to satisfy the entire demand for goggles, it incurs a shortage cost of $40 per unsatisfied unit.

Let’s denote the following variables:

  • X = monthly demand for VR goggles (a normal random variable with mean 160 and standard deviation 40)

  • I = initial inventory level (140 pairs)

  • S = number of goggles sold

  • R = revenue from sales

  • H = inventory holding cost

  • Sh = shortage cost

  • P = monthly profit

The number of goggles sold depends on the demand and the initial inventory level:

S = min(X, I)

The revenue from sales is calculated as:

R = 300S

The inventory holding cost is incurred if there are any unsold goggles at the end of the month:

H = 20(I - S)

The shortage cost occurs if the demand exceeds the initial inventory level:

Sh = 40(max(0, X - I))

Finally, the monthly profit is calculated as:

P = R - H - Sh

To simulate the monthly profit, we can use a spreadsheet simulation model. Here’s an example of how to do this in Excel:

  1. Create a table with the following columns:

    • Month
    • Demand (X)
    • Sold (S)
    • Revenue (R)
    • Holding Cost (H)
    • Shortage Cost (Sh)
    • Profit (P)

  2. In the Demand column, generate 12 random numbers from a normal distribution with mean 160 and standard deviation 40.

  3. In the Sold column, use the MIN function to calculate the minimum of the Demand and Initial Inventory columns.

  4. In the Revenue column, calculate the product of the Sold and Selling Price columns.

  5. In the Holding Cost column, calculate the product of the Unsold Goggles and Holding Cost per Goggle columns.

  6. In the Shortage Cost column, calculate the product of the Unsatisfied Demand and Shortage Cost per Unit columns.

  7. In the Profit column, calculate the difference between the Revenue, Holding Cost, and Shortage Cost columns.

  8. Repeat steps 2-7 for 12 months to get a sample of 12 monthly profits.

  9. Calculate the average of the 12 monthly profits to estimate the average monthly profit under the current policy.

b. Proportion of Months with Satisfied Demand

The proportion of months in which demand is completely satisfied is the proportion of months in which the demand is less than or equal to the initial inventory level.

Proportion of Satisfied Demand = P(X ≤ I)

Since X is a normal random variable with mean 160 and standard deviation 40, we can calculate this probability using the cumulative distribution function (CDF) of the normal distribution:

Proportion of Satisfied Demand = NORM.DIST(140, 160, 40, TRUE)

This formula returns approximately 0.7734, indicating that about 77.34% of the months will have satisfied demand.

c. Profitability Comparison

To compare the profitability of monthly replenishment levels of 140 and 160 pairs of goggles, we can run two separate simulations, one for each replenishment level. For each simulation, we can generate 12 monthly profits and calculate the average monthly profit.

Average Profit at 140 pairs = average(P_140)
Average Profit at 160 pairs = average(P_160)

We can then calculate the difference between the two average profits and construct a 95% confidence interval around this difference.

Difference = Average Profit at 160 pairs - Average Profit at 140 pairs

The 95% confidence interval can be calculated using the t-distribution since we have a sample of 12 observations for each replenishment level.

Confidence Interval = Difference ± t_0.975 *

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